A new algorithm for the maximum likelihood estimation of graphical log-linear models
A \(p\)-dimensional contingency table \(\Omega\) formed by the variables \(V=\{V_{1},\ldots, V_{p}\}\) is the Cartesian product of the sets of indices of the variables in \(V\). A log-linear model \(LL(\Gamma)\) is defined by a class \(\Gamma\) of subsets of \(V\) and it is assumed that \(P(\omega)=\prod_{\gamma\in\Gamma}g_{\gamma}(\omega_{\gamma})\), where \(g_{\gamma}:\Omega_{\gamma}\to R,\;\omega_{\gamma}\in\Omega_{\gamma}\), is a marginal cell of the marginal table defined by the subset of variables \(V_{\gamma},\;\gamma\in\Gamma\). A class \(\Gamma\) which does not contain comparable subsets is called a generating class. A generating class \(\Gamma\) or log-linear model \(LL(\Gamma)\) generated by \(\Gamma\) is called graphical if there are no subsets \(W\subset V\) for which \(| W|\geq 3\) and every \(W'\subset W\) with \(| W'|=| W|-1\) is a generator. Let the first order interaction graph of \(\Gamma\) be denoted by \(G(\Gamma)\). The author proves the following result: Let \(\Gamma\) be a graphical generating class. A distribution \(P\) belongs to \(LL(\Gamma)\) if and only if for every maximal complete subgraph \(W\) of \(\bar G(\Gamma)\) the joint conditional distribution of the variables in \(W\), given all other variables, is independent on \(P\). Then an algorithm to compute maximum likelihood estimates under graphical log-linear models is proposed and a proof of convergence is given.
scientific article; zbMATH DE number 1336643